Paradoxes, Rooms, and Cubes
This week I want to look into more concepts from The Math Book, which is the second book I read for the semester.
The first concept I want to look into more is Newcomb’s Paradox. Here is the paradox: there are two arks. Ark 1 contains $1,000, and Art 2 contains either a spider or the Mona Lisa. There are angels that decide what goes into Art 2, and allegedly they are excellent predictors. Thus, they know what to choose based on what you choose. You are allowed to select either Ark, or both of them. The issue is what you want to choose. You could just choose Ark 1, in which the angels would have predicted you do so, and would have put the Mona Lisa in Ark 2. So would you risk leaving an expensive painting behind? You could just select Ark 2, which the angels would also predict, and they would have put the Mona Lisa in that Ark, but then you’d risk leaving $1,000. Why would anyone risk leaving an extra $1,000? You could select neither, but you’d be an idiot to leave a guaranteed $1,000 behind (and, at least, a pet spider, or, at most, the Mona Lisa). You could also select both, but the angels would predict that you are trying to get the most out of this decision, and place the spider in Ark 2. Now it seems simple, you would probably select just Ark 2 for the Mona Lisa, but the issue comes like this: the angels made their prediction a vast amount of time before you were presented with the choice. This plays serious mind-games with those who make the decision.
A potential solution would be to use a time-machine, but then free-will would not be possible. It’s also a fact that time-machines don’t exist...yet. After pondering this paradox, I’d take Ark 2 only. It’s similar to the prisoners’ problem in game theory, in which choosing the safer option no matter what the opposer would decide to do. IF the angels predict me choosing only Ark 2, then I will have the Mona Lisa. If they have an inaccurate prediction, then they are liars, and I’ll sock them a good one where it counts.
Another concept is Tokarsky’s Unilluminable Room. You are in a room of any size, in any shape, with side passages. Every wall is covered in mirrors. If I am in the room with you, and I light a match, will you be able to see it from anywhere in the room? Tokarsky came up with the concept in 1969, and it perplexed mathematicians for decades. Assuming all walls are straight, Tokarsky found that a 24-sided room could be possible in which the light wouldn’t be seen from certain locations.
Finally, I want to study how to solve a Rubik’s Cube. (I have never tried to actually solve one before).
It’s easy to divide the cube into sides and attempt to solve them one by one. For one side, make a cross so that the adjacent sides have different colored pieces that extend from the cross.
Move the cross to the bottom. Solve each corner of the cross side by rotating the far side upward, then rotate the top to the left, and rotate the first side back down.
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This completes the first side.
Place the completed side on the bottom. Then follow this algorithm to complete the first two levels on the adjacent sides to the completed side.
Rotate top level to the left, rotate far right side up, rotate top level to the right, rotate far right side down, rotate top level to the right, rotate front face counterclockwise once, rotate top level to the left, and front face clockwise once.
The bottom side should be completed, with the first two levels matching on each adjacent side.
Next, match color-sharing colors to their sides. Orient the corners by following this algorithm: rotate far right side up, rotate top level to the left, rotate far right side down, rotate top level left, rotate far right side up, rotate top level left two times, rotate right side down, rotate top level left two more times.
It should look like this:
Next is to permute the edges. If the cube looks like this:
then follow these steps:
Rotate middle column up, top level right, middle column down, top level right twice, middle column up, top level right, and middle column down.
For this look:
Rotate middle column up, top level left, middle column down, top level left twice, middle column up, top level left, and middle column down.
It should look like this:
The final series: rotate right side down, middle level left, right side up twice, middle level right twice, right side down, top level left twice, right side up, middle level right twice, right side down twice, middle level left, right side up, top level left twice, and that’s it! Cube solved.
After trying it a couple times, it started making more sense and became more recognizable. Although, it will take a few more attempts to have it remotely memorized.
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